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Consider the following expression: v = fтВБ (x тАУ rt) + fтВВ (x + rt) where fтВБ and fтВВ represent two traveling waves on a transmission line. In this case
Both waves travel in the positive direction of x
Both waves travel in the negative direction of x
Wave fтВВ travels in the positive direction of x but wave fтВБ travels in the negative direction of x
Wave fтВБ travels in the positive direction of x but wave fтВВ travels in the negative direction of x
Wave fтВБ travels in the positive direction of x but wave fтВВ travels in the negative direction of x
Quick Trick: In the wave equation f(x - vt), a negative sign indicates movement in the positive x-direction, while a positive sign (x + vt) indicates movement in the negative x-direction.
In the wave equation f(x - vt), a negative sign indicates movement in the positive x-direction, while a positive sign (x + vt) indicates movement in the negative x-direction.
Standard Wave Equation Form
A traveling wave is generally expressed as a function of (x +/- vt). If the displacement remains constant, then x +/- vt = constant, meaning x = +/- vt + C.
Analyze Positive Direction
For x = vt + C, x increases as t increases. Thus, the argument (x - rt) represents a wave traveling in the positive x-direction.
Analyze Negative Direction
For x = -vt + C, x decreases as t increases. Thus, the argument (x + rt) represents a wave traveling in the negative x-direction.
A: Incorrect because f2 involves addition, implying negative direction. B: Incorrect because f1 involves subtraction, implying positive direction. C: Incorrect because it reverses the directions for both functions.
D is correct because the mathematical form (x - rt) signifies a delay in signal propagation along the positive axis, whereas (x + rt) signifies propagation along the negative axis.
Whenever you see x +/- vt, treat the sign opposite to the one in the function as the direction of motion: minus is positive direction, plus is negative direction.